I was always told that after lysing the cells, spin them, and only take the supernatant or storage, but why? why do we need to spin the cell lysis to remove the pellet?
The cell pellet contains unbroken cells and debris that (a) might clog a chromatography column and (b) consists of contaminants that you would otherwise have to remove during the purification.
This assumes that you are after something soluble in the cytoplasm. If you want a membrane fraction or inclusion bodies, you may want to keep the pellet and discard the supernatant